Quantum Communication and Quantum Multivariate Polynomial Interpolation

The paper is devoted to the problem of multivariate polynomial interpolation and its application to quantum secret sharing. We show that using quantum Fourier transform one can produce the protocol for quantum secret sharing distribution.Comment: 7 pages, no figure, LaTeX2

Some remarks on the Cegrell's class F\mathcal{F}

In this paper, we study the near-boundary behavior of functions $u\in\mathcal{F}(\Omega)$ in the case where $\Omega$ is strictly pseudoconvex. We also introduce a sufficient condition for belonging to $\mathcal{F}$ in the case where $\Omega$ is the unit ball.Comment: 8 page

Poisson Summation and Endoscopy for SL(3,R)SL(3,\mathbb R)

The group is interesting as the first example of split rank 2 semisimple group, all the irreducible unitary representations of which are known. We make a precise realization of the discrete series representations (in Section 2) by using the Orbit Method and Geometric Quantization, a computation of their traces (Section 3) and an exact formula for the noncommutative Poisson summation and endoscopy of for this group (in Section 4).Comment: arXiv admin note: substantial text overlap with arXiv:1407.690

On a 1D transport equation with nonlocal velocity and supercritical dissipation

We study a 1D transport equation with nonlocal velocity. First, we prove eventual regularization of the viscous regularization when dissipation is in the supercritical range with non-negative initial data. Next, we will prove global regularity for solutions when dissipation is slightly supercritical. Both results utilize a nonlocal maximum principle

Moduli spaces of hyperbolic surfaces and their Weil-Petersson volumes

Moduli spaces of hyperbolic surfaces may be endowed with a symplectic structure via the Weil-Petersson form. Mirzakhani proved that Weil-Petersson volumes exhibit polynomial behaviour and that their coefficients store intersection numbers on moduli spaces of curves. In this survey article, we discuss these results as well as some consequences and applications.Comment: 37 pages - submitted to Handbook of Moduli (edited by G. Farkas and I. Morrison

The asymptotic Weil-Petersson form and intersection theory on M_{g,n}

Moduli spaces of hyperbolic surfaces with geodesic boundary components of fixed lengths may be endowed with a symplectic structure via the Weil-Petersson form. We show that, as the boundary lengths are sent to infinity, the Weil-Petersson form converges to a piecewise linear form first defined by Kontsevich. The proof rests on the observation that a hyperbolic surface with large boundary lengths resembles a graph after appropriately scaling the hyperbolic metric. We also include some applications to intersection theory on moduli spaces of curves.Comment: 22 page

Extending Erd\H{o}s- Beck's theorem to higher dimensions

Erd\H{o}s-Beck theorem states that $n$ points in the plane with at most $n-x$ points collinear define at least $c xn$ lines for some positive constant $c$. In this paper, we will present two ways to extend this result to higher dimensions. Our result has application to point-hyperplane incidences and potential application to the point covering problem

Poisson Summation and Endoscopy for SU(2,1)

In this paper we analyze the endoscopy for $SU(2,1)$. The new results are a precise realization of the discrete series representations (in Section 2), a computation of their traces (Section 3) and an exact formula for the Poisson summation and endoscopy for this group (in Section 4).Comment: Minor misprint changes. arXiv admin note: substantial text overlap with arXiv:1407.681

New SUSY Thought

New SUSY thought is presented. Maximal stop mixing needed for 125 GeV Higgs is linked to the tachyonic stop at the UV boundary. Large $\mu$ does not mean the severe fine tuning if Higgs comes out as a pseudo-Goldstone boson. The small mass of the pseudo-Goldstone Higgs is overcome with extra vector-like fermions needed to explain the Higgs to di-photon rates.Comment: 6 pages, 4 figures, Plenary Talk given at BSM of Particle Physics 2012, Qui Nhon, Vietnam, 15-21 July 201

A simple proof of Zeeman's theorem

It is discussed that Zeeman's theorem can be directly obtained from Liouville's theorem if we assume sufficient differentiability